Question

Find the inverse of y=3 to the power of x +6

Answer 1

Consider the function $y=3^{x+6}$

Interchange the variable $x$ and $y$ to obtain,

$$x=3^{y+6}$$

Take log of base $e$ both sides as,

$$ln( x)=ln( 3^{y+6})$$

Use the power property of logarithm $ln( a^{m})=mln( a)$ to simplify the right hand side of the equation as,

$$ln( x)=(y+6 )ln( 3)$$

$$ln( x)=yln( 3)+6ln( 3)$$

$$ln( x)-6ln( 3)=yln( 3)$$

$$y=\frac{ln( x)-6ln( 3)}{ln( 3)}$$

The sketch of the curve and its inverse is as shown in the figure below:

Here, the curve and it's inverse is symmetric about the line $y=x$ .

Therefore, the inverse of the function $y=3^{x+6}$ is $y=\frac{ln( x)-6ln( 3)}{ln( 3)}$